Q72 / 160 · AP EAMCET 2022 (04 Jul Shift 2) · Mathematics

If $P, Q$ are two points on the curve $y = 2^{x + 2}$ in the rectangular Cartesian coordinate system such that $\bar{O P} \cdot \hat{i} = - 1, \bar{O Q} \cdot \hat{i} = 2$ then $\bar{O Q} - 4 \bar{O P} =$

A
$3 \bar{i} + 8 \bar{j}$
B
$4 \bar{i} + 6 \bar{j}$
Correct answer
C
$6 \bar{i} + 8 \bar{j}$
D
$4 \bar{i} + 3 \bar{j}$

Solution

Given,
${\bar{O P}}_{.} . \hat{i} = - 1 \Rightarrow P_{x} = - 1$
$P$ lies on $y = 2^{x + 2}$
$\Rightarrow y = 2^{- 1 + 2}$
$\Rightarrow y = 2$
So, $(P_{x}, P_{y}) = (- 1, 2)$
Now $\bar{O Q} \cdot \hat{i} = 2 \Rightarrow Q_{x} = 2$
Also $Q$ lies on $y = 2^{x + 2}$
$\Rightarrow y = 2^{2 + 2}$
$\Rightarrow y = 16$
So, $(Q_{x}, Q_{y}) = (2, 16)$
Now $\bar{O Q} = 2 i + 16 j$
And $\bar{O P} = - i + 2 j$
So, $\bar{O Q} - 4 \bar{O P}$ $= 2 i + 16 j + 4 i - 8 j$ $= 6 i + 8 j$

Hence, the correct answer is option: 'C'.

If P,Q are two points on the curve y=2x+2 in the rectangular Cartesian coordinate system such that OP¯·i^=-1, OQ¯·i^=2 then OQ¯-4OP¯=

Solution

Relevant Questions

If $P, Q$ are two points on the curve $y = 2^{x + 2}$ in the rectangular Cartesian coordinate system such that $\bar{O P} \cdot \hat{i} = - 1, \bar{O Q} \cdot \hat{i} = 2$ then $\bar{O Q} - 4 \bar{O P} =$